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This is a question related to a previous question ($a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. Prove that $\sum\limits_{n=1}^\infty a_n/(a_n-1)$ converges?).

This time, I want to give a lower bound on the following infinite sum. $a_n$ is an infinite sequence with $\sum\limits_{n=1}^\infty a_n\leq1$ and $0\leq a_n<1$. W.l.o.g., we sort the $a_n$ in descending order, and set $a_1$ equal to $0 < c < 1$. Is it possible to give a lower bound on the sum in terms of $c$ on $\sum\limits_{n=1}^\infty \frac{a_n}{a_n-1}$?

I tried Taylor series, writing the sum in different forms; Jensen's inequaly is the wrong way around...

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  • $\begingroup$ try using en.wikipedia.org/wiki/… $\endgroup$ – Jneven Mar 15 at 14:47
  • $\begingroup$ $-\frac{1}{1-c}$ will do since $1-a_n \geq 1-c$ and the original sum is positive but $\leq 1$ $\endgroup$ – Conrad Mar 15 at 14:56
  • $\begingroup$ Thank you, I didn't realise that it was that simple! $\endgroup$ – Tchaikovski Mar 15 at 15:10

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